# Chapter 1 One-sample test of proportions

We will start by discussing the one-sample test of proportions. As an example, suppose it has been claimed that among social media users, 73% use Facebook more than once per day, and we wanted to test this claim. Consider the hypotheses

$H_0 : p = 0.73 \text{ versus } H_1 : p \neq 0.73,$

where:

• $$p$$ denotes the proportion of social media users who use Facebook more than once per day
• $$H_0$$ denotes the null hypothesis that the proportion of social media users who use Facebook more than once per day is equal to 0.73 (or as a percentage, 73%)
• $$H_1$$ denotes the alternative hypothesis that the proportion of social media users who use Facebook more than once per day is different from 73%.

In more general terms, suppose we have a random sample of $$n$$ observations with an expected proportion $$p$$ of these observations to have a certain characteristic, letting $$x$$ denote the number of observations in the sample that actually have that characteristic. Equivalently, suppose we conduct $$n$$ independent trials each with probability of success $$p$$ and let $$x$$ denote the number of successes in these $$n$$ trials. Consider the hypotheses

$H_0 : p = p_0\text{ versus } H_1 : p \neq p_0\text{ (or }p<p_0\text{ or }p>p_0),$

where:

• $$p_0$$ denotes the expected proportion under the null hypothesis.

Then, provided $$n$$ is not too small (this will be further discussed shortly), a commonly used statistical test for this type of hypothesis is the one-sample proportion test based on the estimate to $$p$$, which we can denote as $$\hat{p} = x/n$$.

In the example above where we are looking at the proportion of social media users who use Facebook more than once per day, what is the value of $$p_0$$?

0.73

Returning to our example, a survey was carried out to study the social media habits of regular social media users from around the world. Supposing that of the $$n = 484$$ respondents, $$x = 368$$ said they used Facebook more than once per day, we then have that

$\hat{p} = \frac{x}{n} = \frac{368}{484} \approx 0.76.$

Once we know the values of $$x$$ and $$n$$, we then have enough information to calculate $$\hat{p}$$ and then carry out the hypothesis test. Alternatively, if we know the value of $$n$$ and $$\hat{p}$$, we can use this information to calculate $$x$$. Before carrying out the test, it is a good idea to visualise the data and check the assumptions, which we will do in the next section.

### References

Raymond, Mark. 2019. “Social Media Usage Report 2019: User Habits You Need to Know.” 2019. https://www.goodfirms.co/resources/social-media-usage-user-habits-to-know.